Fast rates and strong convergence towards the minimum norm solution of dynamic trajectories with Tikhonov regularization for convex minimization under linear constraints

In this paper, a linear constrained smooth convex minimization problem is considered in a Hilbertian setting. A Tikhonov term is added to obtain a time-dependent strongly convex problem. Then, a primal-dual dynamical system associated to the Lagrangian is presented by adding a time-scale coefficient to the gradient factor. A study of the asymptotic behavior of this dynamical system is presented. First, strong-convergence of solution trajectories to the minimal norm solution occurs without strong convexity of the objective function, then better convergence rate of values, velocity, and constraint toward zero are all achieved.